## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Theoretical and Numerical Aspects of Geometric Variational Problems. Gerd Dziuk, Gerhard Huisken, and John Hutchinson, eds. Proceedings of the Centre for Mathematics and its Applications, v. 26. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1991), Ben - Andrews

### Flow of hypersurfaces by curvature functions

#### Abstract

This seminar concerns a class of flow equations for immersed hypersurfaces, modelled on the well-known mean curvature flow. The flows in this class share much of the qualitative behaviour of the mean curvature flow, but are in general fully nonlinear; this compiicates some parts of their analysis. Other calculations are clarified by the general setting. I will present some results on the behaviour of convex hypersurfaces under these flows, which extend work on specific flows by Huisken (Hul), Tso (Tl) and Chow (Cl-2). Also new is a Harnack inequality for solutions of very general flows; this generalises results of Hamilton (Hal) and Chow (C3). Flows of this kind have some applications in geometry; for such purposes the mean curvature flow is not always the best candidate - I will describe an example which applies to manifolds of non-negative sectional curvature.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416323549

**Mathematical Reviews number (MathSciNet)**

MR1139026

**Zentralblatt MATH identifier**

0758.53028

#### Citation

Andrews, Ben. Flow of hypersurfaces by curvature functions. Theoretical and Numerical Aspects of Geometric Variational Problems, Ben--Andrews, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1991. https://projecteuclid.org/euclid.pcma/1416323549